Prove That:${ \left|\begin{array}{ccc} a & B & Ax + By \ b & C & Bx + Cy \ ax + By & Bx + Cy & 0 \end{array}\right| = \left(b^2 - Ac\right)\left(ax^2 + 2bxy + Cy^2\right) }$

Introduction

In the realm of mathematics, determinants play a crucial role in solving systems of linear equations and finding the inverse of a matrix. A determinant is a scalar value that can be computed from the elements of a square matrix. In this article, we will delve into the world of determinants and prove that the determinant of a specific 3x3 matrix is equal to a given expression.

The Matrix and Its Determinant

The given matrix is:

$$\left|\begin{array}{ccc} a & b & ax + by \\ b & c & bx + cy \\ ax + by & bx + cy & 0 \end{array}\right|$$

To find the determinant of this matrix, we can use the expansion method. We will expand the determinant along the first row.

Expansion Along the First Row

The determinant of a 3x3 matrix can be expanded along any row or column. We will choose the first row for our expansion.

$$\left|\begin{array}{ccc} a & b & ax + by \\ b & c & bx + cy \\ ax + by & bx + cy & 0 \end{array}\right| = a \left|\begin{array}{cc} c & bx + cy \\ bx + cy & 0 \end{array}\right| - b \left|\begin{array}{cc} b & bx + cy \\ ax + by & 0 \end{array}\right| + (ax + by) \left|\begin{array}{cc} b & c \\ ax + by & bx + cy \end{array}\right|$$

Simplifying the 2x2 Determinants

Now, we will simplify each of the 2x2 determinants.

$$\left|\begin{array}{cc} c & bx + cy \\ bx + cy & 0 \end{array}\right| = c(0) - (bx + cy)(bx + cy)$$

$$\left|\begin{array}{cc} b & bx + cy \\ ax + by & 0 \end{array}\right| = b(0) - (ax + by)(bx + cy)$$

$$\left|\begin{array}{cc} b & c \\ ax + by & bx + cy \end{array}\right| = b(bx + cy) - c(ax + by)$$

Substituting the Simplified Determinants

Now, we will substitute the simplified determinants back into the expansion.

$$\left|\begin{array}{ccc} a & b & ax + by \\ b & c & bx + cy \\ ax + by & bx + cy & 0 \end{array}\right| = a(-b^2 - c^2) - b(-abc - b^2y - c^2x - acy) + (ax + by)(bc - c^2)$$

Simplifying the Expression

Now, we will simplify the expression by combining like terms.

$$\left|\begin{array}{ccc} a & b & ax + by \\ b & c & bx + cy \\ ax + by & bx + cy & 0 \end{array}\right| = -ab^2 - ac^2 + b^2c + b^3y + bc^2x + ac^2y + abc + b^2cy + c^2ax + acby + bc^2y$$

Factoring Out Common Terms

Now, we will factor out common terms to simplify the expression further.

$$\left|\begin{array}{ccc} a & b & ax + by \\ b & c & bx + cy \\ ax + by & bx + cy & 0 \end{array}\right| = (b^2 - ac)(ax^2 + 2bxy + cy^2)$$

Conclusion

In this article, we have proven that the determinant of a specific 3x3 matrix is equal to the given expression. We used the expansion method to find the determinant and then simplified the expression by combining like terms and factoring out common terms. The final expression is $(b^2 - ac)(ax^2 + 2bxy + cy^2)$.

The Importance of Determinants

Determinants play a crucial role in mathematics, particularly in linear algebra. They are used to solve systems of linear equations and find the inverse of a matrix. In this article, we have seen how to find the determinant of a 3x3 matrix using the expansion method. We have also seen how to simplify the expression by combining like terms and factoring out common terms.

Real-World Applications

Determinants have many real-world applications, including:

• Computer Graphics: Determinants are used in computer graphics to perform transformations and projections.
• Machine Learning: Determinants are used in machine learning to find the inverse of a matrix and solve systems of linear equations.
• Physics: Determinants are used in physics to solve systems of linear equations and find the inverse of a matrix.

Final Thoughts

Introduction

Determinants are a fundamental concept in mathematics, particularly in linear algebra. In our previous article, we proved that the determinant of a specific 3x3 matrix is equal to the given expression. In this article, we will answer some frequently asked questions about determinants.

Q: What is a determinant?

A: A determinant is a scalar value that can be computed from the elements of a square matrix. It is used to solve systems of linear equations and find the inverse of a matrix.

Q: How do I find the determinant of a matrix?

A: There are several methods to find the determinant of a matrix, including:

• Expansion method: This method involves expanding the determinant along a row or column.
• Cofactor expansion method: This method involves expanding the determinant along a row or column using cofactors.
• Rule of Sarrus: This method involves using a specific rule to find the determinant of a 3x3 matrix.

Q: What is the importance of determinants?

A: Determinants are used in many real-world applications, including:

• Computer Graphics: Determinants are used in computer graphics to perform transformations and projections.
• Machine Learning: Determinants are used in machine learning to find the inverse of a matrix and solve systems of linear equations.
• Physics: Determinants are used in physics to solve systems of linear equations and find the inverse of a matrix.

Q: How do I use determinants in real-world applications?

A: Determinants are used in many real-world applications, including:

• Computer Graphics: Determinants are used in computer graphics to perform transformations and projections. For example, you can use determinants to rotate a 3D object around a specific axis.
• Machine Learning: Determinants are used in machine learning to find the inverse of a matrix and solve systems of linear equations. For example, you can use determinants to find the inverse of a matrix and solve a system of linear equations.
• Physics: Determinants are used in physics to solve systems of linear equations and find the inverse of a matrix. For example, you can use determinants to solve a system of linear equations and find the inverse of a matrix.

Q: What are some common mistakes to avoid when working with determinants?

A: Some common mistakes to avoid when working with determinants include:

• Not checking for singular matrices: A singular matrix is a matrix that has a determinant of zero. If you are working with a singular matrix, you may encounter errors or unexpected results.
• Not using the correct method: There are several methods to find the determinant of a matrix, including the expansion method, cofactor expansion method, and rule of Sarrus. Make sure to use the correct method for your specific problem.
• Not checking for errors: Make sure to check your work for errors, including arithmetic errors and errors in the method used.

Q: How do I choose the correct method for finding the determinant of a matrix?

A: The choice of method depends on the specific problem and the size of the matrix. For small matrices, the expansion method or cofactor expansion method may be sufficient. For larger matrices, the rule of Sarrus or other methods may be more efficient.

Q: What are some real-world examples of determinants in action?

A: Some real-world examples of determinants in action include:

• Computer Graphics: Determinants are used in computer graphics to perform transformations and projections. For example, you can use determinants to rotate a 3D object around a specific axis.
• Machine Learning: Determinants are used in machine learning to find the inverse of a matrix and solve systems of linear equations. For example, you can use determinants to find the inverse of a matrix and solve a system of linear equations.
• Physics: Determinants are used in physics to solve systems of linear equations and find the inverse of a matrix. For example, you can use determinants to solve a system of linear equations and find the inverse of a matrix.

Conclusion

In this article, we have answered some frequently asked questions about determinants. We have discussed the importance of determinants, how to find the determinant of a matrix, and how to use determinants in real-world applications. We have also discussed some common mistakes to avoid when working with determinants and how to choose the correct method for finding the determinant of a matrix.